This is what I have so far, but when I run it, I get a Java mismatch error. This is my array:
char[] letters = {'A', 'B' , 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J'};
/********************************************************************************
    shiftRight() will move the contents of the array one slot to the right
********************************************************************************/
public static void shiftRight( char [] letters )
{
    char last = letters[letters.length-1];          // save off first element
    // shift right
    for( int index =letters.length-1; index >= 0 ; index-- )
        letters[index+1] = letters [index];
    // wrap last element into first slot
    letters[0] = last;
    System.out.print("\nshifted Array: " );
}
                You can do something like:
 public static void shiftRight( char [] letters )
    {
        char last = letters[letters.length-1];          // save off first element
        // shift right
        for( int index =letters.length-2; index >= 0 ; index-- )
            letters[index+1] = letters [index];
        // wrap last element into first slot
        letters[0] = last;
        System.out.print("\nshifted Array: " + Arrays.toString(letters) );
    }
I only modified your: letters.length-1 into letters.length-2 and printed the array.
Another, easier approach is to use, System.arraycopy like:
last = letters[letters.length-1];
System.arraycopy(letters, 0, letters, 1, letters.length-1 );
letters[0] = last;
To print the array you can also use:
System.out.print("{");
for (int i=0;i<letters.length-1;i++)
    System.out.print("'"+letters[i]+",");
System.out.println("'"+letters[letters.length-1]+"'}");
                        When letters[index+1] is executed for the first time in the for-loop (when index = letters.length-1), it's pointing to letters[letters.length] which is not a valid index, since indexes go from 0 to length-1.
Update your for-loop to start index at letters.length-2 and also make sure your array.length>1. That is to say:
   if(letters.length > 1){ //make sure array has minimum two elements
      // shift right
      for( int index = letters.length-2; index >= 0 ; index-- ){
          letters[index+1] = letters [index];
      }
   }
Also in the end, you may print the array as:
   System.out.println("Shifted Array: " +letters); 
EDIT: Sample working code.
If you make the array as Character array  e.g. below in your main (this is required for assistance in printing ONLY)
    Character[] letters = {'A', 'B' , 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J'};
and then pass that to shiftRight method with updated code as below:
  public static void shiftRight( Character [] letters )
  {
    Character last = letters[letters.length-1];  
    if(letters.length >1){ //make sure array has minimum two elements
        // shift right
        for( int index =letters.length-2; index >= 0 ; index-- ){
            letters[index+1] = letters [index];
        }
     }
    letters[0] = last;
    System.out.println(Arrays.toString(letters));
    //^ prints: Shifted Array: [J, A, B, C, D, E, F, G, H, I]
  }
You should be all set.
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